Question

Express the Newton iteration for solving each of the following systems: (a) $$ \begin{array}{r} x_{1}^{2}+x_{1} x_{2}^{3}=9 \\ 3 x_{1}^{2} x_{2}-x_{2}^{3}=4 \end{array} $$ (b) $$ x_{1}+x_{2}-2 x_{1} x_{2}=0 $$ $$ x_{1}^{2}+x_{2}^{2}-2 x_{1}+2 x_{2}=-1 $$ (c) $$ \begin{array}{r} x_{1}^{3}-x_{2}^{2}=0 \\ x_{1}+x_{1}^{2} x_{2}=2 \end{array} $$

Short answer

In summary, we derived the Newton iteration for solving three given systems of equations by following the steps to write the function vector, calculate the Jacobian matrix and express the Newton iteration formula. The Newton iteration formulas for the systems are: (a) $$ \begin{bmatrix} x_{1}^{(k+1)} \\ x_{2}^{(k+1)} \end{bmatrix} = \begin{bmatrix} x_{1}^{(k)} \\ x_{2}^{(k)} \end{bmatrix} - J^{-1}(x^{(k)}) \begin{bmatrix} x_{1}^{2(k)} + x_{1}^{(k)} x_{2}^{3(k)} - 9 \\ 3x_{1}^{2(k)} x_{2}^{(k)} - x_{2}^{3(k)} - 4 \end{bmatrix} $$ (b) $$ \begin{bmatrix} x_{1}^{(k+1)} \\ x_{2}^{(k+1)} \end{bmatrix} = \begin{bmatrix} x_{1}^{(k)} \\ x_{2}^{(k)} \end{bmatrix} - J^{-1}(x^{(k)}) \begin{bmatrix} x_{1}^{(k)} + x_{2}^{(k)} - 2 x_{1}^{(k)} x_{2}^{(k)} \\ {x_{1}^{(k)}}^{2} + {x_{2}^{(k)}}^{2} - 2 x_{1}^{(k)}+2 x_{2}^{(k)} + 1 \end{bmatrix} $$ (c) $$ \begin{bmatrix} x_{1}^{(k+1)} \\ x_{2}^{(k+1)} \end{bmatrix} = \begin{bmatrix} x_{1}^{(k)} \\ x_{2}^{(k)} \end{bmatrix} - J^{-1}(x^{(k)}) \begin{bmatrix} ({x_{1}^{(k)}})^{3} - ({x_{2}^{(k)}})^{2} \\ x_{1}^{(k)} + ({x_{1}^{(k)}})^{2} x_{2}^{(k)} - 2 \end{bmatrix} $$

Step-by-step solution

Write the function vector

For the given system of equations, the function vector F(x) is: $$ F(x) = \begin{bmatrix} x_{1}^{2} + x_{1} x_{2}^{3} - 9 \\ 3x_{1}^{2} x_{2} - x_{2}^{3} - 4 \end{bmatrix} $$

Find the Jacobian matrix

The Jacobian matrix J is the matrix of partial derivatives of the elements of F(x) with respect to the variables x1 and x2: $$ J(x) = \begin{bmatrix} \frac{\partial F_1}{\partial x_1} & \frac{\partial F_1}{\partial x_2} \\ \frac{\partial F_2}{\partial x_1} & \frac{\partial F_2}{\partial x_2} \end{bmatrix} $$ Here, $$ \frac{\partial F_1}{\partial x_1} = 2x_{1} + x_{2}^{3}, \frac{\partial F_1}{\partial x_2} = 3x_{1} x_{2}^{2}\\ \frac{\partial F_2}{\partial x_1} = 6x_{1} x_{2}, \frac{\partial F_2}{\partial x_2} = 3x_{1}^{2} - 3x_{2}^{2} $$ So, the Jacobian matrix is: $$ J(x) = \begin{bmatrix} 2x_{1} + x_{2}^{3} & 3x_{1} x_{2}^{2} \\ 6x_{1} x_{2} & 3x_{1}^{2} - 3x_{2}^{2} \end{bmatrix} $$

Express the Newton iteration formula

For the Newton iteration, we have x(k+1) = x(k) - J^(-1)(x(k)) * F(x(k)): $$ \begin{bmatrix} x_{1}^{(k+1)} \\ x_{2}^{(k+1)} \end{bmatrix} = \begin{bmatrix} x_{1}^{(k)} \\ x_{2}^{(k)} \end{bmatrix} - J^{-1}(x^{(k)}) \begin{bmatrix} x_{1}^{2(k)} + x_{1}^{(k)} x_{2}^{3(k)} - 9 \\ 3x_{1}^{2(k)} x_{2}^{(k)} - x_{2}^{3(k)} - 4 \end{bmatrix} $$ (b)

Write the function vector

For the given system of equations, the function vector F(x) is: $$ F(x) = \begin{bmatrix} x_{1} + x_{2} - 2 x_{1} x_{2} \\ x_{1}^{2} + x_{2}^{2} - 2 x_{1}+2 x_{2} + 1 \end{bmatrix} $$

Find the Jacobian matrix

Following the same procedure as before, we calculate the Jacobian matrix J: $$ J(x) = \begin{bmatrix} 1 - 2x_{2} & 1 - 2x_{1} \\ 2x_{1} - 2 & 2x_{2} + 2 \end{bmatrix} $$

Express the Newton iteration formula

The Newton iteration for this system is: $$ \begin{bmatrix} x_{1}^{(k+1)} \\ x_{2}^{(k+1)} \end{bmatrix} = \begin{bmatrix} x_{1}^{(k)} \\ x_{2}^{(k)} \end{bmatrix} - J^{-1}(x^{(k)}) \begin{bmatrix} x_{1}^{(k)} + x_{2}^{(k)} - 2 x_{1}^{(k)} x_{2}^{(k)} \\ {x_{1}^{(k)}}^{2} + {x_{2}^{(k)}}^{2} - 2 x_{1}^{(k)}+2 x_{2}^{(k)} + 1 \end{bmatrix} $$ (c)

Write the function vector

For the given system of equations, the function vector F(x) is: $$ F(x)= \begin{bmatrix} x_{1}^{3} - x_{2}^{2} \\ x_{1} + x_{1}^{2} x_{2} - 2 \end{bmatrix} $$

Find the Jacobian matrix

Following the procedure, we calculate the Jacobian matrix J: $$ J(x) = \begin{bmatrix} 3x_{1}^{2} & -2x_{2} \\ 1 + 2x_{1}x_{2} & x_{1}^{2} \end{bmatrix} $$

Express the Newton iteration formula

The Newton iteration for this system is: $$ \begin{bmatrix} x_{1}^{(k+1)} \\ x_{2}^{(k+1)} \end{bmatrix} = \begin{bmatrix} x_{1}^{(k)} \\ x_{2}^{(k)} \end{bmatrix} - J^{-1}(x^{(k)}) \begin{bmatrix} ({x_{1}^{(k)}})^{3} - ({x_{2}^{(k)}})^{2} \\ x_{1}^{(k)} + ({x_{1}^{(k)}})^{2} x_{2}^{(k)} - 2 \end{bmatrix} $$

Key concepts

Jacobian Matrix

The Jacobian matrix is a core element in numerical analysis, particularly in solving systems of nonlinear equations using methods like Newton's Method. It is essentially a matrix of first-order partial derivatives. For functions involving multiple variables, the Jacobian is a compact way of expressing how the output of the functions changes as the input variables change.

In our exercise, when dealing with a set of functions where each function depends on multiple variables, the Jacobian matrix helps us understand the sensitivity of each function to changes in each variable. Here's how it typically looks:

• The rows of the Jacobian represent the partial derivatives of each function with respect to each variable.
• The columns represent the partial derivatives of all functions with respect to a single variable.

For example, in Exercise (a), the Jacobian matrix,\[J(x) = \begin{bmatrix}2x_{1} + x_{2}^{3} & 3x_{1} x_{2}^{2} \6x_{1} x_{2} & 3x_{1}^{2} - 3x_{2}^{2}\end{bmatrix}\]illustrates how changes in the variables \(x_1\) and \(x_2\) affect the system's equations. Calculating the Jacobian is the first step towards using Newton's Method for solving these equations.

Nonlinear Systems

Nonlinear systems of equations consist of one or more equations in which the unknown or the variables appear to a power other than one (or have other non-linearities like products of variables). Solving such systems is more challenging than linear systems because they cannot be reduced to a simple matrix operation.

For the given exercise, we have multiple systems that need solving, each containing nonlinear relationships between the variables. Take Exercise (c), where the system is:

• \(x_1^3 - x_2^2 = 0\)
• \(x_1 + x_1^2x_2 = 2\)

These systems are nonlinear because they involve variables raised to a power greater than one, or involve multiplication of variables. The nonlinearity of these equations is what requires advanced techniques like Newton's Method for finding solutions. Nonlinear systems often have multiple solutions, and finding them involves iteration, using initial guesses and refining them through a process based on the behavior of the functions.

Numerical Analysis

Numerical analysis is a field of mathematics that develops and analyzes algorithms for approximating solutions to mathematical problems, many of which cannot be solved exactly. This is particularly important for solving equations that are too complex for simple algebraic solutions.

Newton's Method, as applied in this exercise, is a numerical technique for finding successively better approximations to the roots (or zeroes) of a real-valued function. This method is iterative, meaning it improves estimates step by step until a sufficient level of accuracy is achieved.

The key steps involve:

• Starting with an initial guess for the roots.
• Using the Jacobian matrix to understand how the system of functions behaves.
• Correcting the guess using the Newton iteration formula:

\[x^{(k+1)} = x^{(k)} - J^{-1}(x^{(k)}) \cdot F(x^{(k)})\]This formula updates the guess \(x^{(k)}\) using the inverse of the Jacobian and the function vector \(F(x)\) evaluated at \(x^{(k)}\). Numerical analysis helps in making these estimates converge quickly to an accurate solution, instead of calculating them directly, which might be computationally exhaustive or impossible.